In this paper, we propose a new method for constructing supersaturated designs that is based on the Kronecker product of two carefully-chosen matrices. The construction method leads to a partitioning of the columns of the design such that the columns within a group are correlated to the others within the same group, but are orthogonal to any factor in any other group. We refer to the resulting designs as group orthogonal supersaturated designs (GO-SSDs). We leverage this group structure to obtain an unbiased estimate of the error variance, and to develop an effective, design-based model selection procedure. Simulation results show that the use of these designs, in conjunction with our model selection procedure enables the identification of larger numbers of active main effects than have previously been reported for supersaturated designs. The designs can also be used in group screening; however, unlike previous group-screening procedures, with our designs, main effects in a group are not confounded.